Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34632 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2031 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 2031 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,28,22,20,29,27)(18,23,26,31,25,24,32,30);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20)(18,24)(19,22)(21,27)(23,25)(26,30)(28,29)(31,32);(3,8)(5,6)(11,15)(13,16)(18,25)(23,24)(26,32)(30,31)Polycyclic presentation:
2^9|b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298903);